# Interdisciplinary Applied Mathematics

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2. The error between the full-order and the reduced-order models should be as small as possible.

3.    The reduced-order model should preserve the essential properties of the full-order system.

Therefore, the reduced-order linear system should be of the form

C„z (t) + Gnz(t) = Bnu(t),    y(t) = Ln z(t),    (17.10)

where z(t) 6 Ж» is a state vector, u(t) G Rm is the input excitation vector, and y(t) G is the output vector. Here Cn, Gn G Rnxn are system matrices, Bn G Rnxm and Ln G Rnxp are the input and output distribution arrays, respectively; n is the state space dimension, which should be much smaller than N. Assuming a single-input single-output (SISO) system for simplicity, p = m = 1. In this case, we use b and l for input and output vectors, respectively. The MIMO system can be dealt with in a similar manner.

The Krylov subspace technique (Srinivasan et al., 2001) reduces the original system (equation (17.9)) to the reduced system (equation (17.10)). Before we discuss the reduction method, it is important to understand the concept of Krylov subspaces. A Krylov subspace is a subspace spanned by a    sequence of vectors    generated    by    a    given    matrix    and a    vector    as    fol

lows. Given a matrix A and a starting vector r, the nth Krylov subspace Kn(A, r) is spanned by a sequence of n column vectors:

Kn(A, r) = span[r, Ar, A2r, A3r,…, An-1r].

This is called the right Krylov subspace. When A is asymmetric, there exists    a    left    Krylov    subspace generated    by    AT    and a    starting    vector    l

defined by

Kn (AT, l) = span[l, AT l, (AT )21,…, (AT )n-1l].

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